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Eigenvector equation

  1. Oct 17, 2008 #1
    Hi, this is actually for my general relativity class, but I thought I would get more help in the math section of the forums, since it involves very little physics, or even not at all.

    1. The problem statement, all variables and given/known data
    Take Tab and Sab to be the covariant components of two tensors. Consider the determinant equation for [itex]\lambda[/itex] :

    | [tex]\lambda[/tex]Tab - Sab |= 0

    Prove that the roots of this equation are scalars, making clear what you mean by scalar.

    2. Relevant equations



    3. The attempt at a solution
    Well If I solve for the determinant I think I should get a quartic equation for the eigenvalues [itex]\lambda[/itex] of the form
    [tex]\lambda[/tex]^4 + a1[tex]\lambda[/tex]^3 + a2[tex]\lambda[/tex]^2 + a3[tex]\lambda[/tex] + a4 = 0
    Or not? Will I get an equation involving the components of the tensors T and S??
    I just want to make sure I am understanding the question and I'm headed in the right path.
    Any suggestions are greatly appreciated.
     
  2. jcsd
  3. Oct 17, 2008 #2
    Maybe you need to show that the eigenvalues are dependent on T_ab and S_ab in such a way that makes them invariant under transformations to another reference frame. Is that how "scalar" is defined in GR?

    There's probably a clever way to answer the question that won't involve writing out the equations in detail.
     
  4. Oct 20, 2008 #3
    I just don't know where to start. Do you suggest getting the determinant of the matrix and equaling that to 0? That will take so long. Is there some theorem or something? Anybody know? :S
     
  5. Oct 20, 2008 #4
    Sorry I don't think I know enough to help. I'm only studying special relativity, so I don't know how to interpret this determinant equation.
     
  6. Oct 21, 2008 #5
    Yeah this is a pretty weird problem. One of my classmates is helping me now :)
    Thanks anyway
     
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